Enhancing Network Security in Distributed Systems Using Middle Roman Dominating Functions

A Middle Roman dominating function (MRDN) on a graph G = (V,E) is a function f:v→{0,1,2,3} satisfying the condition that every vertex u with f(u)=0 is adjacent to at most one vertex v with f(v)=2 or 3. Further if a vertex is assigned 2, then at most two of its vertices can be assigned 0 and if a vertex is assigned 3, then all its neighbours can be assigned 0. The weight of a MRDF is the value f(V(G))=∑_uϵV▒〖f(u〗). The Middle Roman domination number γ_MR (G) is the minimum weight of a MRDF on G. In this paper, we introduce Middle Roman Domination number denoted as γ_MR (G), study the properties of the function, present some characterization and determine γ_MR (G)-value for some graphs. Middle Roman Dominating Functions (MRDFs) present a unique approach within graph theory, with significant implications for various fields in computer science. By assigning values to vertices under specific constraints, MRDFs enable the optimization of resources, enhancing network security, load balancing, and energy efficiency in distributed systems. This paper explores the application of MRDFs in scenarios such as intrusion detection, resilient network design, task scheduling, and sensor activation. By minimizing the overall weight while maintaining functional requirements, MRDFs provide an effective strategy for addressing challenges in network topology, resource allocation, and fault tolerance. The versatility of MRDFs makes them a valuable tool in the development of robust and efficient computing systems.